Partial Regularity for Singular Solutions to the Monge‐Ampère Equation | Zendy

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Partial Regularity for Singular Solutions to the Monge‐Ampère Equation

Author(s) -

Mooney Connor

Publication year - 2015

Publication title -

communications on pure and applied mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 3.12

H-Index - 115

eISSN - 1097-0312

pISSN - 0010-3640

DOI - 10.1002/cpa.21534

Subject(s) - mathematics , monge–ampère equation , hausdorff measure , hausdorff dimension , bounded function , regular polygon , hausdorff space , pure mathematics , continuation , mathematical analysis , zero (linguistics) , set (abstract data type) , combinatorics , geometry , linguistics , philosophy , computer science , programming language

We prove that solutions to the Monge‐Ampère inequalitydet D 2 u ≥ 1in ℝ n are strictly convex away from a singular set of Hausdorff ( n ‐1)‐dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to det D 2 u = 1 with singular set of Hausdorff dimension as close as we like to n ‐1. As a consequence we obtain W 2,1 regularity for the Monge‐Ampère equation with bounded right‐hand side and unique continuation for the Monge‐Ampère equation with sufficiently regular right‐hand side. © 2015 Wiley Periodicals, Inc.

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